.

National Renewable Energy Laboratory

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303-275-3000 • www.nrel.gov

Operated for the U.S. Department of Energy 

Office of Energy Efficiency and Renewable Energy 

by Midwest Research Institute • Battelle 

Contract No. DE-AC36-99-GO10337 

J.A. del Cueto 

Prepared for the 31st IEEE Photovoltaics Specialists 

Conference and Exhibition 

Lake Buena Vista, Florida 

January 3–7, 2005 

February 2005 • NREL/CP-520-36565 

Closed-Form Solutions and 

Parameterization of the Problem 

of Current-Voltage Performance 

of Polycrystalline Photovoltaic 

Modules Deployed at Fixed 

Latitude Tilt 
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CLOSED-FORM SOLUTIONS AND PARAMETERIZATION OF THE PROBLEM OF 

CURRENT-VOLTAGE PERFORMANCE OF POLYCRYSTALLINE PHOTOVOLTAIC 

MODULES DEPLOYED AT FIXED LATITUDE TILT 

J.A. del Cueto, National Renewable Energy Laboratory (NREL), 1617 Cole Boulevard, Golden, CO 80401, USA 

ABSTRACT 

One obstacle to wider market penetration for thin-film 

polycrystalline photovoltaic (PV) modules is the lack of 

field performance data, notably temperature-coefficient 

data for the current-voltage (I-V) characteristics over all 

irradiance levels encountered in the field. Generally, temperature coefficient and performance data are available 

only in a restricted range of illumination around 1-sun intensity. In this paper, the I-V performance data are presented, analyzed, and parameterized across a wide range 

of illumination levels and temperatures, allowing the modeling of the performance for three polycrystalline PV technologies: cadmium telluride, copper indium diselenide, and 

polycrystalline silicon. The data are scrutinized for clearsky and diffuse-illumination conditions. 

INTRODUCTION 

The prevalence of crystalline and polycrystalline silicon technologies (bulk-Si) may be ascribed to many factors that translate to overall comfort of PV integrators with 

bulk-Si technology. With thin films, in contrast, there is 

considerable unease in the industry for reasons including: 

lower efficiency, past failures, module-reliability issues, 

plus a general lack of field performance data and a dearth 

of temperature-coefficient data for thin-film modules [1]. 

Currently existing module-performance data, even for 

bulk-Si, tend to be quite limited, being available typically in 

a restricted range of illumination near 1-sun. Furthermore, 

even much of the data presented at lower irradiance levels 

are generally obtained by attenuating the simulator intensity [2], which ignores angle-of-incidence or spectral effects present at lower illumination [3]; most likely, this 

thereby renders an unrealistic simulation of how lower light 

intensities affect the I-V parameters. In this paper, the I-V 

performance for polycrystalline thin-film PV modules is 

parameterized based on outdoor measurements, and a set 

of closed-form parameters and formulae are presented 

from which these can be translated to a wide variation of 

conditions. 

METHOD 

Outdoor performance data for three distinct PV technologies are scrutinized: three cadmium telluride (CdTe) 

modules (A, B, C); two copper indium diselenide (CIS) 

modules (A, B), and two polycrystalline silicon (pc-Si) 

modules (A, B). Their I-V characteristics are analyzed: the 

behavior of the open-circuit voltage (Voc), short-circuit current (Isc), fill factor (FF), and their respective temperature 

coefficients are derived from outdoor measurements, for 

irradiance levels ranging from low (50 W/m2

) to high (1200 

W/m2

), with module temperatures ranging 20°–65°C. Irradiance levels were measured with a broadband 

pyranometer mounted in the same plane-of-array and angle as the modules; the temperatures represent back-ofmodule measurements; both are sampled simultaneously

with the I-V traces. The featured modules are deployed 

facing south at fixed latitude tilt on the Performance and 

Energy Ratings Testbed (PERT) on the roof of the Outdoor Test Facility at NREL. The scrutinized data were obtained from 4 years worth of records, representing 30,000 

to 50,000 I-V traces for each module. The Voc, Isc, and FF

data, plus their temperature derivatives, were subsequently fit, evaluated at 25°C, and parameterized as polynomial functions of irradiance. Additionally, the data were 

segregated for clear-sky and diffuse illumination conditions 

using a statistical filter and model. All the parameters 

needed to calculate module performance for a wide range 

of operating conditions are presented. 

One approach to modeling module performance is to 

interpolate the I-V trace at arbitrary irradiance in terms of 

traces already measured at nearby values [4]. However, 

this involves some recursive evaluation, does not represent a closed-form approach, and it may not be clear 

which are the best traces to use. Another approach is to 

extend the definition of the temperature coefficients—

defined as the derivative with respect to temperature of the 

I-V parameter in question divided by its value at standard 

test conditions (STC)—to cover that of the derivatives 

measured at other irradiance, with these divided by the 

parameter’s value at the new irradiance and 25°C. Note 

that even though the temperature derivatives of the I-V 

parameters might be slowly varying functions of irradiance, 

when these are divided by the respective parameter values at 25°C at the same irradiance, the resulting coefficient is convoluted with complex irradiance dependence. 

Hence, one problem that is encountered in expressing 

module performance in closed-form is formulating the dependence of the I-V parameters themselves. 

In its simplest form, a PV device is modeled via the 

diode equation with parasitic series- and shunt-resistance 

terms. Deriving irradiance and temperature dependence 

for the I-V parameters (Isc, Voc, and FF) is fairly complex 

when parasitic resistances are taken into account, the 

details of which calculations can be found in the literature 

[5]. This reference points out the dependence of the FF

on the reciprocal Voc, which translates to a logarithmic 

dependence with irradiance. In this analysis, the I-V parameters and their temperature derivatives were reduced 

and evaluated at 25°C module temperature across varying 

1
irradiance in 50 W/m2

-wide discrete intervals thereof. The 

data are subsequently fit to a polynomial function in powers of irradiance, or its logarithm. One to three such terms 

were employed in the fits, containing 17–23 evaluations 

spanning irradiance values 50–1200 W/m2

. The number of 

fitting terms was limited to three, with the criteria for keeping terms based on statistical significance using the FDistribution test at the 99% level of confidence. 

RESULTS 

Voc[25,Irr]/Voc(STC) (%)

100

56

60

64

68

72

76

80

84

88

92

96

Irradiance (W/m2)

0 200 400 600 800 1000 1200

Beta [%/°C]

CIS A CIS B

CdTe A CdTe B

CdTe C pc-Si A

pc-Si B

 Diffuse Illumination

Clear-Sky Conditions

Clear-Sky Conditions

-0.48

-0.44

-0.40

-0.36

-0.32

-0.28

-0.24

-0.20

-0.16

-0.12

-0.08

-0.04

0.00

-0.48

-0.44

-0.40

-0.36

-0.32

-0.28

-0.24

-0.20

-0.16

-0.12

-0.08

-0.04

0.00

Beta [%/°C]

Fig. 1. Voc data at 25°C and varying irradiance relative to 

values at STC at bottom, and beta values at middle, plotted against irradiance for 7 modules, for clear-sky conditions. Beta values at top are for diffuse illumination data. 

Figure 1 depicts the ratio of module Voc at 25°C 

across varying illumination as ratios to values at STC and 

its temperature coefficients, beta, respectively, in bottom 

and middle portions of the graph, plotted against irradiance along the ordinate axis, for predominantly clear-sky

data. At top, the beta for diffuse illumination is depicted. 

Beta is defined as the derivative of Voc with respect to 

temperature evaluated at a given irradiance, divided by the 

value Voc at 25°C and the same irradiance. At 1000 W/m2

, 

beta values range from –0.28% to –0.20% per °C, –0.28% 

to –0.40% per °C, and –0.35% per °C, respectively, for 

CdTe, CIS, and pc-Si modules. The values of beta for 

clear-sky data become larger, more negative—about 25% 

larger—toward lower irradiance at 200 W/m2

. Thereafter, 

they appear to increase toward positive direction while still 

remaining negative down to 50 W/m2

 irradiance. There are 

slight differences between clear-sky and diffuse data, except at the lowest irradiance. 

Alpha [Irr] (%/°C) Alpha [Irr] (%/°C)

Irradiance (W/m2)

0 200 400 600 800 1000 1200

pc-Si A

CIS B

pc-Si B

CdTe A

CIS A

CdTe C

CdTe B

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Clear-Sky Illumination

Diffuse-Cloudy Illumination

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

Fig. 2. Temperature coefficients of Isc, alpha, plotted vs. 

irradiance for predominantly diffuse (top) and clear (bottom) conditions for seven polycrystalline PV modules. 

In Fig. 2, the coefficients alpha are depicted for predominantly diffuse- and clear-sky illumination conditions, 

respectively, at top and bottom portions of the graph, plotted against irradiance along the ordinate axis. Alpha is 

defined as the derivative with respect to temperature of Isc

at a given irradiance divided by Isc at 25°C and the same 

irradiance. If instead, alpha were defined as the derivative 

divided by the value of Isc at STC, then the alpha values at 

low irradiance do not exhibit the increase depicted in Fig. 

2. A striking feature of the data in Fig. 2 is the substantial 

difference between the two data sets at low illumination, 

below 250 W/m2

 irradiance: for diffuse illumination, the 

data do not exhibit as pronounced an increase as for the 

clear-sky data. Otherwise, above 300 W/m2

 irradiance, the 

alpha values between sets are quite similar, except for one 

of the CdTe modules (CdTe C). Also note that for the CIS 

modules under some irradiance conditions, the alpha values become negative. This behavior was observed for this 

module when measured under the Spire simulator. 

Figure 3 portrays the FF data at 25°C under varying illumination, as ratios of their values at STC, and its temperature coefficients, respectively, in bottom and top portions of the graph, plotted against irradiance, for predominantly clear-sky conditions. Except for CdTe A, all the FF

data vary smoothly in going from high to low illumination—

typically exhibiting slight increases initially as the irradiance decreases, and then decreasing toward low irradiance. This behavior is consistent with series-resistance 

losses at high irradiance that become smaller at lower 

irradiance; and then by the functional dependence with 

Voc, series and shunt resistance at low irradiance [5]. With 

the one exception (CdTe A), the temperature coefficients 

of the FF show monotonically decreasing values with increasing irradiance, varying from small positive values to 

negative values in the range 200–500 W/m2

. For diffuse 

illumination data, the FF and temperature coefficients are 

similar to those depicted in Fig. 3, except for CdTe A. 

2
FF [25,Irr] / FF [STC] (%)

58

62

66

70

74

78

82

86

90

94

98

102

106

110

CIS A CIS B

CdTe A CdTe B

CdTe C pc-Si A

pc-Si B

Irradiance (W/m2)

0 200 400 600 800 1000 1200

T-Coeff. FF [Irr] (% /°C)

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

Fig. 3. FF ratios at 25°C relative to values at STC at bottom and its temperature coefficients at top, plotted against 

irradiance for seven modules, for clear-sky conditions. 

For CdTe A, the FF behavior below 600W/m2

 irradiance is complex, but likely to be real as each point represents 500 to 700 I-V traces. In contrast, for diffuse illumination, the FF and temperature coefficients do not show

the complex behavior seen at low irradiance in Fig. 3. Instead, the FF values increase at low irradiance and reach 

110% of the value at STC at ~100 W/m2

. Its temperature 

coefficients obey a monotonic increase in values with irradiance, exhibiting nearly identical values to those shown 

in Fig. 3 for irradiance above 600 W/m2

. 

ANALYSIS 

Parametric representations for Voc, Isc, FF data plus 

their temperature coefficients beta, alpha, and Tcoeff FF

are tabulated in the next section: Tables 1–6 and 7–12, 

respectively, for clear-sky and diffuse lighting. In these 

tables, the first, second, and third columns, respectively, 

denote the module, respective parameter value at STC, 

and one standard deviation of the fit. For Voc, Isc, and FF,

the fourth and higher columns represent the coefficients of 

the terms (ci

) expressed as ratios of their STC values. For 

the temperature coefficients, the fourth and higher columns are simply the values of the coefficients of the fitting 

terms. The specific functional terms in the irradiance used 

are listed along the top row of each column, where the 

fourth column is the constant term, the fifth is the logarithm, the sixth is the linear, etc., as terms of the irradiance 

expressed in W/m2

. All the values of the temperature coefficients are evaluated as % change per °C with respect to 

the parameter evaluated at 25°C, at the same irradiance. 

To calculate any I-V parameter, P(x), or its temperature 

coefficient (Tcoeff(x)) at arbitrary irradiance, use Eqn. 1, 

with PSTC values taken from the second columns for Voc, 

Isc, and FF, and with PSTC

 = 1 for the temperature coefficients; and with x = irradiance expressed in units of W/m2

. 

Subsequently, to calculate the values of Voc, Isc, or FF at 

realistic module temperature T, multiply the P(x) by the 

temperature coefficient times the difference (T-25°C) as 

per Eqn. 2. The Voc, Isc, and FF are formulated such that 

their respective values at STC multiply a unitless function 

of irradiance inside the braces in Eqn. 1, which could accommodate similar modules simply by using their values 

at STC, rather than the ones studied. 

( ) { ( ) / / } (1) 2

1 2

3

3

2

0 1 2 P x P c c Ln x c x c x c x c x c x L

STC + + + + + − + − = ⋅

P(x,T) = P(x)⋅{1+Tcoeff (x)⋅[T − 25]} (2)

More elaborate parameterization terms are used to 

represent Isc due to complex behavior in low-illumination 

situations, with differences between clear-sky and diffuse 

lighting. For clear-sky data, the fit contains log and quadratic terms—not usually associated with the formulation of 

Isc—whereas for diffuse data, these terms are missing 

(except for CdTe A). The source of the complexity lies in 

how low-illumination situations arise. For clear-sky conditions, this occurs when the angle-of-incidence (AOI) between the sun and the modules’ surfaces is large and the 

direct-beam component becomes increasingly reflected 

and less absorbed. Diffuse illumination occurs either from 

obscuration by clouds or from lighting coming chiefly from 

the sky dome. It is instructive to analyze the effect that 

AOI has on Isc via the photoresponse, which is defined as 

the short-circuit current density divided by the irradiance—

a quantity that is independent of area with units of mA/W. 

These data are depicted in Fig. 5 plotted vs. AOI for six of 

the modules, along with the average irradiance at those 

AOI. When Isc is strictly linear with irradiance, the photoresponse is constant: this is borne out at AOI < 60°. With 

increasing AOI, the direct-beam component is reflected 

away more rapidly than the pyranometer’s response drops 

off, with the net effect that the modules do not absorb this 

measurable energy and the photoresponse decreases. 

Above ~84° AOI, more of the irradiance under clear skies 

becomes diffuse and behaves as if it were coming from 

the sky dome at an average AOI ~45°, thus becoming less 

reflected, increasing the photoresponse, and convoluting 

the function for Isc. 

100

140

180

220

260

300

0 10 20 30 40 50 60 70 80 90 100

AOI

Irrad. (W/m2)

0

200

400

600

800

1000

CIS B

pc-Si A

CdTe B

CIS A

CdTe A

CdTe C

Photorespon. (

mA/W)

Fig. 5 Photoresponse at 25°C and irradiance plotted vs. 

AOI, respectively, at bottom and top for clear-sky data. 

3
PARAMETRIC REPRESENTATION 

 Table 1. Voc Parametric Representation [25°C, x = irradiance], Clear-Sky Conditions. 

Module Voc

STC

 (V) σ (V) c0 / Voc

STC cL/ VocSTC

 : Ln(x) c1 / Voc

STC

 : x1 c2 / Voc

STC

 : x2 c3 / Voc

STC

 : x3

CIS A 25.468 0.0471 0.4194 0.08658 -1.744E-11

CIS B 54.142 0.1243 0.3441 0.10491 -6.885E-05

CdTe A 86.408 0.1811 0.6463 0.05770 -4.482E-05

CdTe B 88.585 0.3716 0.2752 0.11882 -9.597E-05

CdTe C 40.389 0.1337 -0.2375 0.20602 -4.159E-07 2.303E-10

pc-Si A 21.487 0.0216 0.5730 0.06379 -1.366E-08 

pc-Si B 21.369 0.0262 0.5768 0.06348 -1.528E-08 

 Table 2. Isc Parametric Representation [25°C, x = irradiance], Clear-Sky Conditions. 

Module Isc

STC

 (A) σ (Α) c0 / Isc

STC cL / IscSTC

 : Ln(x) c1 / Isc

STC

: x1 c2 /Isc

STC

: x2

CIS A 2.3449 0.0090 0.2564 -0.070144 1.461E-03 -2.326E-07 

CIS B 2.0024 0.0068 0.2367 -0.065320 1.432E-03 -2.173E-07 

CdTe A 0.9074 0.0033 0.2797 -0.076025 1.513E-03 -2.678E-07 

CdTe B 0.8858 0.0024 0.2018 -0.055979 1.371E-03 -1.865E-07 

CdTe C 2.8345 0.0137 0.1374 -0.036841 1.188E-03 -7.067E-08 

pc-Si A 2.1995 0.0081 0.2389 -0.065585 1.468E-03 -2.542E-07 

pc-Si B 2.1960 0.0081 0.2390 -0.066055 1.476E-03 -2.589E-07 

Table 3. Fill Factor (FF) Parametric Representation [ 25°C, x = irradiance], Clear-Sky Conditions. 

Module FFSTC

(%) σ (%) c0/FFSTC cL/FFSTC:Ln(x) c1/FFSTC:x c2/FFSTC: x2 c3/FFSTC: x3 c -1/FFSTC: 1/x c -2/FFSTC: 1/x2

CIS A 62.00 0.18 1.584 -0.07967 -34.673 872.63

CIS B 70.66 0.32 -0.221 0.21258 -2.481E-04 182.80

CdTe A 57.27 1.11 1.039 2.659E-05 -6.597E-11

CdTe B 52.71 0.40 1.881 -0.11980 -55.036 1181.59

CdTe C 56.97 0.26 -0.225 0.18449 -5.436E-08 4.906

pc-Si A 73.82 0.18 0.524 0.08792 -1.314E-04

pc-Si B 74.34 0.15 0.517 0.08917 -1.328E-04

 Table 4. Beta Temperature-Coefficient Parametric Representation [ 25°C, x = irradiance] , Clear-Sky Conditions. 

Module betaSTC (%/°C) σ (%/°C) c0 cL: Ln(x) C1: x1 c2: x2 c3: x3 c -1:1/x

CIS A -3.671E-03 7.5348 E-05 -5.0413E-03 1.298E-06 0.07233

CIS B -2.754E-03 6.4270 E-05 -4.4477E-03 2.991E-06 -1.474E-09 0.17667

CdTe A -2.457E-03 4.4782 E-05 -5.4159E-03 5.648E-06 -2.689E-09

CdTe B -2.879E-03 8.8631 E-05 -5.8722E-03 5.915E-06 -3.086E-09 0.16383

CdTe C -2.117E-03 1.9973 E-04 5.1144E-03 -1.820E-03 8.945E-06 -3.603E-09

pc-Si A -3.266E-03 3.5330 E-05 -3.8843E-03 1.379E-09 -8.198E-13 0.05860

pc-Si B -3.349E-03 3.8321 E-05 -4.3989E-03 1.731E-06 -7.463E-10 0.06513

 Table 5. Alpha Temperature-Coefficient Parametric Representation [25°C, x = irradiance], Clear-Sky Conditions. 

Module alphaSTC (%/°C) σ (%/°C) c0 cL: Ln(x) c -1:1/x c -2 :1/x2

CIS A 8.915E-05 4.5082 E-04 -5.2416E-02 7.071E-03 3.767 -106.21 

CIS B 6.840E-04 5.4406 E-04 -4.4044E-02 6.087E-03 2.749 -67.06 

CdTe A 1.226E-03 6.6722 E-04 -4.3573E-02 6.050E-03 3.093 -85.62 

CdTe B 1.400E-03 6.2247 E-04 -4.1258E-02 5.742E-03 3.079 -84.66 

CdTe C 1.387E-03 7.5610 E-04 7.5033E-04 0.654 -17.38

pc-Si A 1.094E-03 2.2339 E-04 -4.0156E-02 5.468E-03 3.582 -105.03 

pc-Si B 1.153E-03 5.4992 E-04 -3.4441E-02 4.789E-03 2.582 -66.09 

 

Table 6. FF Temperature-Coefficient Parametric Representation [25°C, x = irradiance], Clear-Sky Conditions. 

Module Tcoeff-FFSTC (%/°C) σ (%) c0 cL: Ln(x) c1: x1 c2: x2 c3: x3 c -1 : 1/x c -2: 1/x2

CIS A -8.282E-04 1.02E-04 2.517E-03 -4.508E-04 -2.313E-10

CIS B -1.079E-03 2.21E-04 -1.502E-03 2.455E-07 0.1777

CdTe A 1.925E-03 6.01E-04 2.792E-02 -7.386E-03 4.504E-05 -2.001E-08

CdTe B -4.031E-04 4.09E-04 -5.568E-05 -5.186E-13 0.1711

CdTe C -1.361E-03 3.81E-04 6.266E-04 -1.987E-06

pc-Si A -1.795E-03 8.31E-05 2.124E-03 -5.869E-04 0.1404 -4.893

pc-Si B -1.727E-03 1.04E-04 9.449E-04 -4.255E-04 0.2784 -10.504

Table 7. Voc Parametric Representation [25°C, x = irradiance] , Diffuse-Illumination Conditions. 

Module Voc

STC

 (V) σ (V) c0 / Voc

STC cL/ VocSTC

: Ln(x) c1 / Voc

STC

 : x1 c-1 / Voc

STC

 : 1/x c-2 / Voc

STC

: 1/x2

CIS A 25.514 0.0570 0.1271 0.02597 -1.095E-05

CIS B 54.038 0.1053 0.7044 0.04360 -5.616

CdTe A 86.319 0.1295 0.6749 0.05225 -3.584E-05

CdTe B 88.423 0.2620 0.7458 0.03789 -7.533

CdTe C 40.420 0.1031 -0.0135 0.17166 -1.721E-04 -186.13

pc-Si A 21.482 0.0183 0.7024 0.04340 -2.192

pc-Si B 21.376 0.0204 0.7007 0.04362 -2.057

4
Table 8. Isc Parametric Representation [25°C, x = irradiance], Diffuse-Illumination Conditions. 

Module Voc

STC

 (V) σ (V) c0 / Voc

STC cL/ VocSTC

 : Ln(x) c1 / Voc

STC

 : x1 c3 / Voc

STC

 : x3

CIS A 2.3322 0.0106 0.00236 1.012E-03 -1.472E-11 

CIS B 2.0032 0.0106 -0.00146 1.015E-03 -1.365E-11 

CdTe A 0.9126 0.0038 0.07805 -0.018616 1.099E-03 -4.841E-11 

CdTe B 0.8923 0.0058 0.00350 9.965E-04

CdTe C 2.8777 0.0151 0.00267 9.973E-04

pc-Si A 2.2028 0.0116 0.00685 1.009E-03 -1.616E-11 

pc-Si B 2.2029 0.0090 0.00352 1.017E-03 -2.047E-11 

Table 9. Fill Factor Parametric Representation [25°C, x = irradiance], Diffuse-Illumination Conditions. 

Module FFSTC

 (%) σ (%) c0 / FFSTC cL/ FFSTC: Ln(x) c1/ FFSTC: x c2/ FFSTC : x2 c-1/ FFSTC : 1/x

CIS A 61.66 0.24 0.632 0.0838 -2.562E-04 4.551E-08

CIS B 70.74 0.21 0.071 0.1595 -1.725E-04

CdTe A 57.16 0.31 1.148 -1.446E-04 -3.155

CdTe B 52.46 0.47 1.122 -1.071E-04 -15.039

CdTe C 59.74 0.38 0.107 0.1172 8.340E-05

pc-Si A 73.40 0.49 0.691 0.0600 -1.029E-04 -2.384 

pc-Si B 74.26 0.26 0.556 0.0827 -1.270E-04 -0.563 

Table 10.Beta Temperature-Coefficient Parametric Representation [25°C, x = irradiance], Diffuse-Illumination Conditions. 

Module betaSTC (%/°C) σ (%/°C) C0 cL: Ln(x) c1: x c2: x2 c-1: 1/x c -2:1/x2

CIS A -3.792E-03 6.593E-05 -8.029E-03 0.000608 0.03566 

CIS B -2.708E-03 4.794E-05 -3.196E-03 4.866E-07 1.38084

CdTe A -2.488E-03 8.467E-05 -7.330E-03 0.000701

CdTe B -2.773E-03 1.339E-04 -4.688E-03 3.736E-06 -1.820E-09

CdTe C -1.942E-03 2.218E-04 -2.006E-02 0.002481 1.00600 -28.00201

pc-Si A -3.286E-03 3.397E-05 -5.646E-03 0.000337 0.03227 

pc-Si B -3.371E-03 3.740E-05 -5.732E-03 0.000338 0.02556

Table 11.Alpha Temperature-Coefficient Parametric Representation [25°C, x=irradiance], Diffuse-Illumination Conditions. 

Module alphaSTC (%/°C) σ (%/°C) c0 cL: Ln(x) c 1: x c -1:1/x c -2 :1/x2

CIS A -4.781E-05 2.027E-04 8.976E-03 -0.001684 2.607E-06

CIS B 3.668E-04 2.484E-04 -1.141E-02 0.001584 0.85674 -21.293 

CdTe A 8.208E-04 1.870E-04 -1.223E-02 0.001718 1.22348 -40.083 

CdTe B 6.931E-04 3.241E-04 1.186E-02 -0.002038 2.906E-06

CdTe C 1.764E-03 3.099E-04 4.330E-02 -0.007749 1.245E-05 -0.46201

pc-Si A 9.495E-04 3.680E-04 6.832E-04 0.27144 -5.089 

pc-Si B 9.496E-04 1.736E-04 -5.150E-03 0.000806 0.54502 -12.441 

Table 12. FF Temperature-Coefficient Parametric Representation [25°C, x = irradiance], Diffuse-Illumination Conditions. 

Module Tcoeff-FFSTC (%/°C) σ (%/°C) c0 cL: Ln(x) c1: x1 c2: x2 c3: x3 c -1 :1/x

CIS A -5.924E-04 2.153E-04 -5.924E-04

CIS B -1.172E-03 1.256E-04 4.583E-04 -0.000241 0.03215

CdTe A 1.567E-03 2.306E-04 -1.148E-04 2.332E-06 -6.503E-13 

CdTe B -2.516E-06 3.641E-04 1.016E-04 -1.758E-13 0.07165

CdTe C -2.991E-03 4.820E-04 1.714E-03 -4.706E-06

pc-Si A -1.732E-03 2.878E-04 3.299E-03 -0.000793 4.489E-07

pc-Si B -1.812E-03 2.486E-04 -9.084E-05 -3.848E-06 2.127E-09

CONCLUSIONS 

Parametric representations for Voc, Isc, FF, and their 

temperature coefficients were presented in closed form for 

seven polycrystalline PV modules, from which these three 

primary I-V parameters and their temperature coefficients 

may be calculated for irradiance ranging from 50 W/m2 to 

1200 W/m2

, and for module temperatures ranging from 

20°–65°C, for predominantly clear-sky and diffuseillumination conditions. 

ACKNOWLEDGEMENTS 

This work was supported by the U.S. Department of 

Energy under Contract No. DE-AC36-99GO10337. 

REFERENCES

[1] B. Hammond, “Thin-Film PV Technologies: A utility

perspective,” 3rd National Thin-Film Module Reliability

Team Meeting, Albuquerque NM, Sept. 2003. 

[2] B. Marion, B. Kroposki, K. Emery, J. del Cueto, D. 

Myers, and C. Osterwald, “Validation of a photovoltaic 

module energy ratings procedure at NREL,” NREL 

Technical Report # 520-26909, August 1999. 

[3] R. Rüther, G. Kleiss, and K. Reichi, Solar Energy Materials and Solar Cells, 71, 2002, pp. 375–385. 

[4] B. Marion, S. Rummel, and A. Anderberg, Progress in 

PV: Research and Applications, 12, 2004, pp. 1–16. 

[5] de Vos, Solar Cells 8, 1983, pp. 283–296. 

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4. TITLE AND SUBTITLE

Closed-Form Solutions and Parameterization of the Problem of 

Current-Voltage Performance of Polycrystalline Photovoltaic 

Modules Deployed at Fixed Latitude Tilt 

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One obstacle to wider market penetration for thin-film polycrystalline photovoltaic (PV) modules is the lack of field performance 

data, notably temperature-coefficient data for the current-voltage (I-V) characteristics over all irradiance levels encountered in the 

field. Generally, temperature coefficient and performance data are available only in a restricted range of illumination around 1-sun 

intensity. In this paper, the I-V performance data are presented, analyzed, and parameterized across a wide range of illumination 

levels and temperatures, allowing the modeling of the performance for three polycrystalline PV technologies: cadmium telluride,

copper indium diselenide, and polycrystalline silicon. The data are scrutinized for clear-sky and diffuse-illumination conditions. 

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PV; thin-film; current-voltage (I-V); 1-sun intensity; cadmium telluride, copper indium diselenide; polycrystalline silicon; diffuseillumination; 

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